Question

A man walks a distance of 3 units from the origin towards the north-east $\left(N{45}^{o}E\right)$ direction. Form there, he walks a distance of 4 units towards the north-west $\left(N{45}^{o}W\right)$ direction to reach a point P. Then the position of P in the Argand plane is

• A. $3e^{i\pi /4}+4i$
• B. $(3-4i)e^{i\pi /4}$
• C. $(4-3i)e^{i\pi /4}$
• D. $(3+4i)e^{i\pi /4}$

Answer 1

The correct option is D $(3+4i)e^{i\pi /4}$

After walking $3$ units from origin towards north-east, the man reaches position $A$.

$A$ is $3e^{i\pi /4}$ in Argand plane.

After walking $4$ units in the north-west direction from $A$ he reaches $P$. Let $P$ be $z$ in Argand plane.

$AO\perp AP$

$\Rightarrow \frac{0-3e^{i\pi /4}}{z-3e^{i\pi /4}}=\frac{3}{4}(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})=\frac{3i}{4}$

$\Rightarrow$ $\frac{-3e^{i\pi /4}}{z-3e^{i\pi /4}}=\frac{3i}{4}$

$\Rightarrow$ $-4e^{i\pi /4}=i(z-3e^{i\pi /4})$

$z=3e^{i\pi /4}+4i{e}^{i\pi /4}=(3+4i)e^{i\pi /4}$